Rack and Quandle Representations and Connections to the g-digroup Structure

En este trabajo estudiamos algunas propiedades algebraicas de las estructuras de rack y quandle así como la teoría de representaciones de estos objetos. Concretamente, demostramos que existe una correspondencia entre las representaciones fuertes e irreducibles de un rack finito y conexo con las repr...

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Autores:
Vallejos Cifuentes, Ricardo Esteban
Tipo de recurso:
Fecha de publicación:
2023
Institución:
Universidad Nacional de Colombia
Repositorio:
Universidad Nacional de Colombia
Idioma:
eng
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oai:repositorio.unal.edu.co:unal/85625
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https://repositorio.unal.edu.co/handle/unal/85625
https://repositorio.unal.edu.co/
Palabra clave:
510 - Matemáticas::512 - Álgebra
Teoría de los grupos
Grupos finitos
Representación de grupos (Matemáticas)
Algebra - Enseñanza
Racks
Quandles
Rack representations
Enveloping group
Representaciones de racks
Rights
openAccess
License
Reconocimiento 4.0 Internacional
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oai_identifier_str oai:repositorio.unal.edu.co:unal/85625
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dc.title.eng.fl_str_mv Rack and Quandle Representations and Connections to the g-digroup Structure
dc.title.translated.spa.fl_str_mv Representaciones de Racks y Quandles y conexiones con los g-digrupos
title Rack and Quandle Representations and Connections to the g-digroup Structure
spellingShingle Rack and Quandle Representations and Connections to the g-digroup Structure
510 - Matemáticas::512 - Álgebra
Teoría de los grupos
Grupos finitos
Representación de grupos (Matemáticas)
Algebra - Enseñanza
Racks
Quandles
Rack representations
Enveloping group
Representaciones de racks
title_short Rack and Quandle Representations and Connections to the g-digroup Structure
title_full Rack and Quandle Representations and Connections to the g-digroup Structure
title_fullStr Rack and Quandle Representations and Connections to the g-digroup Structure
title_full_unstemmed Rack and Quandle Representations and Connections to the g-digroup Structure
title_sort Rack and Quandle Representations and Connections to the g-digroup Structure
dc.creator.fl_str_mv Vallejos Cifuentes, Ricardo Esteban
dc.contributor.advisor.none.fl_str_mv Rodriguez Nieto, José Gregorio
dc.contributor.author.none.fl_str_mv Vallejos Cifuentes, Ricardo Esteban
dc.contributor.orcid.spa.fl_str_mv Vallejos Cifuentes, Ricardo Esteban [0009-0000-4216-0473]
dc.subject.ddc.spa.fl_str_mv 510 - Matemáticas::512 - Álgebra
topic 510 - Matemáticas::512 - Álgebra
Teoría de los grupos
Grupos finitos
Representación de grupos (Matemáticas)
Algebra - Enseñanza
Racks
Quandles
Rack representations
Enveloping group
Representaciones de racks
dc.subject.lemb.none.fl_str_mv Teoría de los grupos
Grupos finitos
Representación de grupos (Matemáticas)
Algebra - Enseñanza
dc.subject.proposal.none.fl_str_mv Racks
Quandles
dc.subject.proposal.eng.fl_str_mv Rack representations
Enveloping group
dc.subject.proposal.spa.fl_str_mv Representaciones de racks
description En este trabajo estudiamos algunas propiedades algebraicas de las estructuras de rack y quandle así como la teoría de representaciones de estos objetos. Concretamente, demostramos que existe una correspondencia entre las representaciones fuertes e irreducibles de un rack finito y conexo con las representaciones irreducibles de su grupo finito envolvente, lo cual implica que podemos estudiar las representaciones fuertes de un rack finito y conexo a través de la teoría de representaciones de grupos finitos. Por último, estudiamos la estructura de digrupo generalizado y su relación con la estructura de rack. (Tomado de la fuente)
publishDate 2023
dc.date.issued.none.fl_str_mv 2023-12-05
dc.date.accessioned.none.fl_str_mv 2024-02-05T20:34:01Z
dc.date.available.none.fl_str_mv 2024-02-05T20:34:01Z
dc.type.spa.fl_str_mv Trabajo de grado - Maestría
dc.type.driver.spa.fl_str_mv info:eu-repo/semantics/masterThesis
dc.type.version.spa.fl_str_mv info:eu-repo/semantics/acceptedVersion
dc.type.content.spa.fl_str_mv Text
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status_str acceptedVersion
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dc.identifier.instname.spa.fl_str_mv Universidad Nacional de Colombia
dc.identifier.reponame.spa.fl_str_mv Repositorio Institucional Universidad Nacional de Colombia
dc.identifier.repourl.spa.fl_str_mv https://repositorio.unal.edu.co/
url https://repositorio.unal.edu.co/handle/unal/85625
https://repositorio.unal.edu.co/
identifier_str_mv Universidad Nacional de Colombia
Repositorio Institucional Universidad Nacional de Colombia
dc.language.iso.spa.fl_str_mv eng
language eng
dc.relation.indexed.spa.fl_str_mv LaReferencia
dc.relation.references.spa.fl_str_mv Nicolás Andruskiewitsch and Matías Graña. From racks to pointed hopf algebras. Advances in mathematics, 178(2):177–243, 2003.
Valeriy Bardakov, Inder Bir Singh Passi, and Mahender Singh. Quandle rings. Journal of Algebra and its applications, 18(8):1950157, 2019.
Mohamed Elhamdadi, Jennifer Macquarrie, and Ricardo Restrepo. Automorphism groups of quandles. Journal of Algebra and its Applications, 11(1):1250008, 2012.
Mohamed Elhamdadi and El-ka ̈ıoum Moutuou. Finitely stable racks and rack representations. Communications in Algebra, 46(11):4787–4802, 2018.
Roger Fenn and Colin Rourke. Racks and links in codimension two. Journal of Knot Theory and Its Ramifications, 1(4):343–406, 1992.
Matias Graña, Istv án Heckenberger, and Leandro Vendramin. Nichols algebras of group type with many quadratic relations. Advances in mathematics, 227(5):1956–1989, 2011.
David Joyce. An algebraic approach to symmetry with applications to knot theory. Phd thesis, University of Pennsylvania, 1979.
David Joyce. A classifying invariant of knots, the knot quandle. Journal of Pure and Applied Algebra, 23(1):37–65, 1982.
Michael Kinyon. Leibniz algebras, lie racks, and digroups. Journal of Lie Theory, 17:99–114, 2007.
Victoria Lebed and Leandro Vendramin. On structure groups of set-theoretic solutions to the yang–baxter equation. Proceedings of the Edinburgh Mathematical Society, 62(3):683–717, 2019.
Sergei Vladimirovich Matveev. Distributive groupoids in knot theory. Mathematics of the USSR-Sbornik, 47(1):78–88, 1982.
Takefumi Nosaka. Homotopical interpretation of link invariants from finite quandles. Topology and its applications, 193:1–30, 2015.
Takefumi Nosaka. Quandles and topological pairs: symmetry,knots and cohomology. Springer, 2017.
Jos é Gregorio Rodríguez, Olga Patricia Salazar, and Raúl Velásquez. The structure of g- digroup actions and representation theory. Algebra and Discrete Mathematics, 32(1):103–126, 2021.
Olga Patricia Salazar, Ra ́ul Vel ́asquez, and L.A Wills. Generalized digroups. Communications in Algebra, 44(7):2760–2785, 2016.
Markus Szymik. Permutations, power operations, and the center of the category of racks. Communications in Algebra, 46(1):230–240, 2018.
Mituhisa Takasaki. Abstraction of symmetric transformations. Tohoku Mathematical Journal, 49:145–207, 1943.
Leandro Vendramin. On the classification of quandles of low order. Journal of Knot Theory and Its Ramifications, 21(09):1250088, 2012.
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dc.rights.license.spa.fl_str_mv Reconocimiento 4.0 Internacional
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dc.format.extent.spa.fl_str_mv 66 páginas
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dc.publisher.spa.fl_str_mv Universidad Nacional de Colombia
dc.publisher.program.spa.fl_str_mv Medellín - Ciencias - Maestría en Ciencias - Matemáticas
dc.publisher.faculty.spa.fl_str_mv Facultad de Ciencias
dc.publisher.place.spa.fl_str_mv Medellín
dc.publisher.branch.spa.fl_str_mv Universidad Nacional de Colombia - Sede Medellín
institution Universidad Nacional de Colombia
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spelling Reconocimiento 4.0 Internacionalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2Rodriguez Nieto, José Gregorio4b6bb8c89efe5778cb8d5d05065b938eVallejos Cifuentes, Ricardo Esteban583fb1f8e7a365e88c9d1c84ada13198Vallejos Cifuentes, Ricardo Esteban [0009-0000-4216-0473]2024-02-05T20:34:01Z2024-02-05T20:34:01Z2023-12-05https://repositorio.unal.edu.co/handle/unal/85625Universidad Nacional de ColombiaRepositorio Institucional Universidad Nacional de Colombiahttps://repositorio.unal.edu.co/En este trabajo estudiamos algunas propiedades algebraicas de las estructuras de rack y quandle así como la teoría de representaciones de estos objetos. Concretamente, demostramos que existe una correspondencia entre las representaciones fuertes e irreducibles de un rack finito y conexo con las representaciones irreducibles de su grupo finito envolvente, lo cual implica que podemos estudiar las representaciones fuertes de un rack finito y conexo a través de la teoría de representaciones de grupos finitos. Por último, estudiamos la estructura de digrupo generalizado y su relación con la estructura de rack. (Tomado de la fuente)In this research we study some algebraic properties of the rack and quandle structure as well as the representation theory of these objects. We establish a correspondence between the irreducible strong representations of a finite, connected rack with the irreducible representation of its finite enveloping group, which implies that the study of strong representations of a finite, connected rack can be approached through the representation theory of finite groups. Finally, we study the g-digroup structure and its connection to the rack structure.MaestríaMagíster en Ciencias - MatemáticasÁlgebraMaestría en Ciencias - Matemáticas66 páginasapplication/pdfengUniversidad Nacional de ColombiaMedellín - Ciencias - Maestría en Ciencias - MatemáticasFacultad de CienciasMedellínUniversidad Nacional de Colombia - Sede Medellín510 - Matemáticas::512 - ÁlgebraTeoría de los gruposGrupos finitosRepresentación de grupos (Matemáticas)Algebra - EnseñanzaRacksQuandlesRack representationsEnveloping groupRepresentaciones de racksRack and Quandle Representations and Connections to the g-digroup StructureRepresentaciones de Racks y Quandles y conexiones con los g-digruposTrabajo de grado - Maestríainfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/acceptedVersionTexthttp://purl.org/redcol/resource_type/TMLaReferenciaNicolás Andruskiewitsch and Matías Graña. From racks to pointed hopf algebras. Advances in mathematics, 178(2):177–243, 2003.Valeriy Bardakov, Inder Bir Singh Passi, and Mahender Singh. Quandle rings. Journal of Algebra and its applications, 18(8):1950157, 2019.Mohamed Elhamdadi, Jennifer Macquarrie, and Ricardo Restrepo. Automorphism groups of quandles. Journal of Algebra and its Applications, 11(1):1250008, 2012.Mohamed Elhamdadi and El-ka ̈ıoum Moutuou. Finitely stable racks and rack representations. Communications in Algebra, 46(11):4787–4802, 2018.Roger Fenn and Colin Rourke. Racks and links in codimension two. Journal of Knot Theory and Its Ramifications, 1(4):343–406, 1992.Matias Graña, Istv án Heckenberger, and Leandro Vendramin. Nichols algebras of group type with many quadratic relations. Advances in mathematics, 227(5):1956–1989, 2011.David Joyce. An algebraic approach to symmetry with applications to knot theory. Phd thesis, University of Pennsylvania, 1979.David Joyce. A classifying invariant of knots, the knot quandle. Journal of Pure and Applied Algebra, 23(1):37–65, 1982.Michael Kinyon. Leibniz algebras, lie racks, and digroups. Journal of Lie Theory, 17:99–114, 2007.Victoria Lebed and Leandro Vendramin. On structure groups of set-theoretic solutions to the yang–baxter equation. Proceedings of the Edinburgh Mathematical Society, 62(3):683–717, 2019.Sergei Vladimirovich Matveev. Distributive groupoids in knot theory. Mathematics of the USSR-Sbornik, 47(1):78–88, 1982.Takefumi Nosaka. Homotopical interpretation of link invariants from finite quandles. Topology and its applications, 193:1–30, 2015.Takefumi Nosaka. Quandles and topological pairs: symmetry,knots and cohomology. Springer, 2017.Jos é Gregorio Rodríguez, Olga Patricia Salazar, and Raúl Velásquez. The structure of g- digroup actions and representation theory. Algebra and Discrete Mathematics, 32(1):103–126, 2021.Olga Patricia Salazar, Ra ́ul Vel ́asquez, and L.A Wills. Generalized digroups. Communications in Algebra, 44(7):2760–2785, 2016.Markus Szymik. Permutations, power operations, and the center of the category of racks. Communications in Algebra, 46(1):230–240, 2018.Mituhisa Takasaki. Abstraction of symmetric transformations. Tohoku Mathematical Journal, 49:145–207, 1943.Leandro Vendramin. On the classification of quandles of low order. Journal of Knot Theory and Its Ramifications, 21(09):1250088, 2012.InvestigadoresLICENSElicense.txtlicense.txttext/plain; charset=utf-85879https://repositorio.unal.edu.co/bitstream/unal/85625/1/license.txteb34b1cf90b7e1103fc9dfd26be24b4aMD51ORIGINAL1085313710.2023.pdf1085313710.2023.pdfTesis de maestría en Ciencias - Matemáticasapplication/pdf790447https://repositorio.unal.edu.co/bitstream/unal/85625/2/1085313710.2023.pdf04db4683a14f3cd4cfebd799268854a6MD52THUMBNAIL1085313710.2023.pdf.jpg1085313710.2023.pdf.jpgGenerated Thumbnailimage/jpeg4292https://repositorio.unal.edu.co/bitstream/unal/85625/3/1085313710.2023.pdf.jpg19daadcb3a264440349217de9f572cb1MD53unal/85625oai:repositorio.unal.edu.co:unal/856252024-02-05 23:03:44.457Repositorio Institucional Universidad Nacional de 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